In both X-ray computed tomography (CT) and positron emission tomography (PET), iterative reconstruction can be used to generate reconstructed images of a subject. In both cases a system matrix A can be used to express relations between the detected X-ray data y and the reconstructed image x, as expressed by the system-matrix equationAx=y. In CT image reconstruction, the system matrix A represents X-ray trajectories through a subject, such that the X-ray intensity after propagating through the subject is dependent upon the attenuation along the X-ray trajectory through the subject. In PET, the system matrix A represents the line of response (LOR) indicated by the detection of two X-rays from a single electron-positron annihilation event. The LOR represents a line between the two detectors that detect the two X-rays from the annihilation event.
While there are various iterative reconstruction (IR) methods, such as the algebraic reconstruction technique, one common IR method is to solve the optimization problem
            arg      ⁢                          ⁢      min        x    ⁢      {                                                  x            -            y                                    W        2            +              β        ⁢                                  ⁢                  U          ⁡                      (            x            )                                }  to obtain the argument x that minimizes the above cost function in the curly brackets. For example, in X-ray CT, A is the system matrix that represents X-ray trajectories (e.g., a line integral or Radon transform or X-ray transform representing a given X-ray path) from an X-ray source through an object OBJ to an X-ray detector, y represents projection images (e.g., the X-ray intensity at the X-ray detector) generated at a series of projection angles, and x represents the reconstructed image of the X-ray attenuation of the object OBJ. The notation ∥g∥w2 signifies a weighted inner product of the form ½gTWg, wherein W is the weight matrix. For example, when the noise between X-ray detector elements is uncorrelated, the weight matrix W can be the inverse of the noise of the projection data. The system-matrix term ∥Ax−y∥w2 is minimized when the forward projection A of the reconstructed image x provides a good approximation to all measured projection images y. In the above expression, U(x) is a regularization function, and β is a regularization parameter that weights the relative contributions of the system-matrix term and the regularization term.
IR methods augmented with regularization can have several advantages over other reconstruction methods such as filtered back-projection. For example, IR methods augmented with regularization can produce high-quality reconstructions even when the projection data includes only sparse projection angles (i.e., few-view projection data) or when the signal-to-noise ratio is poor (i.e., noisy projection data). For few-view, limited-angle, and noisy projection data, the use of a regularization function imposes predefined characteristics on the reconstructed image according to some a priori model of the object OBJ. For example, enforcing positivity for the attenuation coefficients can provide regularization based on the a priori model that each pixel either absorbs X-rays or is transparent (i.e., no X-ray gain), which, as a practical matter, is a virtual certainty in clinical applications. Note the word “pixel” designates a value in an array of arbitrary dimension, including, e.g., two-dimensional pixels in a two-dimensional image array, and three-dimensional volume pixels or voxels in a three-dimensional image array.
Other regularization terms can similarly rely on a priori knowledge of characteristics or constraints imposed on the reconstructed image. For example, minimizing the “total variation” (TV) in conjunction with projection on convex sets (POCS) is also a very popular regularization scheme. The TV-minimization algorithm assumes that the image is predominantly uniform over large regions with sharp transitions at the boundaries of the uniform regions, which is generally true for a clinical image of various organs, each of which will exhibit an approximately constant X-ray absorption coefficient throughout the organ (e.g., bone can have a first absorption coefficient, the lungs have a second coefficient, and the heart has a third coefficient). When the a priori model corresponds well to the image object OBJ, these regularized IR algorithms can produce good image quality even though the reconstruction problem is significantly underdetermined (e.g., few view scenarios), missing projection angles, or noisy.
While the regularization term can generally improve the noise characteristics of a reconstructed image x, the size of the improvement to the signal-to-noise ratio in a reconstructed image will depend on the size of the regularization parameter β. When β is large, the signal-to-noise ratio of the reconstructed image can decrease, and when β is small the signal-to-noise ratio of the reconstructed image can increase. On the other hand, making the regularization parameter β very large can hinder the spatial resolution for certain choices of regularization functions.
Additionally, the signal-to-noise ratio of the reconstructed image can depend on the size of the display field of view (dFOV), the system model expressed by the forward-projection and back-projection matrices, and statistical properties of the projection data. Current methods are insufficient to select a regularization parameter β, prior to performing regularized IR, that is ensured to generate a reconstructed image having specified statistical characteristics for a given set of projection data, system model, and dFOV. This limitation of the current methods for selecting the regularization parameter β for a certain reconstruction task and to achieve a reconstructed image having specified noise properties and image quality applies to both CT and PET systems.
PET systems are widely used in medical imaging. In positron emission tomography (PET) imaging, a radiopharmaceutical agent is introduced into the object to be imaged via injection, inhalation, or ingestion. After administration of the radiopharmaceutical, the physical and bio-molecular properties of the agent cause it to concentrate at specific locations in the human body. The actual spatial distribution of the agent, the intensity of the region of accumulation of the agent, and the kinetics of the process from administration to its eventual elimination are all factors that may have clinical significance. During this process, a positron emitter attached to the radiopharmaceutical agent will emit positrons according to the physical properties of the isotope, such as half-life, branching ratio, etc.
The radionuclide emits positrons, and when an emitted positron collides with an electron, an annihilation event occurs, wherein the positron and electron are combined. Most of the time, an annihilation event produces two gamma rays (at 511 keV) traveling at substantially 180 degrees apart.
In order to be able to reconstruct the spatio-temporal distribution of the radio-isotope via tomographic reconstruction principles, each detected event will need to be characterized for its energy (i.e., amount of light generated), its location, and its timing. By detecting the two gamma rays, and drawing a line between their locations, i.e., the line-of-response (LOR), one can determine the likely location of the original disintegration. While this process will only identify a line of possible interaction, by accumulating a large number of those lines, and through a tomographic reconstruction process, the original distribution can be estimated. In addition to the location of the two scintillation events, if accurate timing (within a few hundred picoseconds) is available, a time-of-flight (TOF) calculation can add more information regarding the likely position of the event along the line. Limitations in the timing resolution of the scanner will determine the accuracy of the positioning along this line. The collection of a large number of events creates the necessary information for an image of an object to be estimated through tomographic reconstruction.
PET imaging systems use detectors positioned across from one another to detect the gamma rays emitting from the object. A ring of detectors can be used in order to detect gamma rays coming from each angle. Thus, a PET scanner can be substantially cylindrical to be able to capture as much radiation as possible, which should be, by definition, isotropic. PET scanners can be composed of several thousand individual crystals (i.e., scintillator elements), which are arranged in two-dimensional scintillator arrays that are packaged in modules with photodetectors to measure the light pulses from respective scintillation events. The relative pulse energy measured by the photodetectors is used to identify the position of the scintillation event.
Computed tomography (CT) systems have some similarities and some differences with PET systems. CT systems and methods are also widely used for medical imaging and diagnosis. CT systems generally create images of one or more sectional slices through a subject's body. A radiation source, such as an X-ray source, irradiates the body from one side. A collimator, generally adjacent to the X-ray source, limits the angular extent of the X-ray beam, so that radiation impinging on the body is substantially confined to a planar (or volume) region defining a cross-sectional slice of the body. At least one detector (and generally many more than one detector) on the opposite side of the body receives radiation transmitted through the body substantially in the plane of the slice. The attenuation of the radiation that has passed through the body is measured by processing electrical signals received from the detector. The projection data at a series of projection angles (i.e., a CT scan) are then recorded and/or processed to reconstruct an image of the sampled planar region (or volume).
A CT sinogram is an arrangement of the projection data displaying attenuation through the body as a function of “space” along a detector array along a first axis (or set of axes) and “time/angle” of a CT scan along a second axis. The space dimension refers to the position along a one-dimensional array of X-ray detectors. The time/angle dimension refers to the projection angle of X-rays changing as a function of time, such that as time progresses the projection angle increments and projection measurements are performed at a linear succession of projection angles. The attenuation resulting from a particular volume will trace out a sine wave around the vertical axis—volumes farther from the axis of rotation having sine waves with larger amplitudes, the phase of a sine wave determining the volume's angular position around the rotation axis. Performing an inverse Radon transform or equivalent image reconstruction method reconstructs an image from the projection data in the sinogram—the reconstructed image corresponding to a cross-sectional slice (or volume) of the body.